SU(2) Yang–Mills Theory: Waves, Particles, and Quantum Thermodynamics

• Main Author: Ralf Hofmann
• Format: Article
• Language: English
• Published: MDPI AG 2016-08-01
• Series: Entropy
• Subjects: Harrington–Shepard caloron; (anti)selfduality; electric and magnetic dipole densities; vacuum permittivity and permeability; Poincaré group; quantum of action; Boltzmann weight; Bose–Einstein distribution function
• Online Access: http://www.mdpi.com/1099-4300/18/9/310

We elucidate how Quantum Thermodynamics at temperature T emerges from pure and classical S U ( 2 ) Yang–Mills theory on a four-dimensional Euclidean spacetime slice S 1 × R 3 . The concept of a (deconfining) thermal ground state, composed of certain solutions to the fundamental, classical Yang–Mills equation, allows for a unified addressation of both (classical) wave- and (quantum) particle-like excitations thereof. More definitely, the thermal ground state represents the interplay between nonpropagating, periodic configurations which are electric-magnetically (anti)selfdual in a non-trivial way and possess topological charge modulus unity. Their trivial-holonomy versions—Harrington–Shepard (HS) (anti)calorons—yield an accurate a priori estimate of the thermal ground state in terms of spatially coarse-grained centers, each containing one quantum of action ℏ localized at its inmost spacetime point, which induce an inert adjoint scalar field ϕ ( | ϕ | spatio-temporally constant). The field ϕ , in turn, implies an effective pure-gauge configuration, a μ gs , accurately describing HS (anti)caloron overlap. Spatial homogeneity of the thermal ground-state estimate ϕ , a μ gs demands that (anti)caloron centers are densely packed, thus representing a collective departure from (anti)selfduality. Effectively, such a “nervous” microscopic situation gives rise to two static phenomena: finite ground-state energy density ρ gs and pressure P gs with ρ gs = − P gs as well as the (adjoint) Higgs mechanism. The peripheries of HS (anti)calorons are static and resemble (anti)selfdual dipole fields whose apparent dipole moments are determined by | ϕ | and T, protecting them against deformation potentially caused by overlap. Such a protection extends to the spatial density of HS (anti)caloron centers. Thus the vacuum electric permittivity ϵ 0 and magnetic permeability μ 0 , supporting the propagation of wave-like disturbances in the U ( 1 ) Cartan subalgebra of S U ( 2 ) , can be reliably calculated for disturbances which do not probe HS (anti)caloron centers. Both ϵ 0 and μ 0 turn out to be temperature independent in thermal equilibrium but also for an isolated, monochromatic U ( 1 ) wave. HS (anti)caloron centers, on the other hand, react onto wave-like disturbances, which would resolve their spatio-temporal structure, by indeterministic emissions of quanta of energy and momentum. Thermodynamically seen, such events are Boltzmann weighted and occur independently at distinct locations in space and instants in (Minkowskian) time, entailing the Bose–Einstein distribution. Small correlative ramifications associate with effective radiative corrections, e.g., in terms of polarization tensors. We comment on an S U ( 2 ) × S U ( 2 ) based gauge-theory model, describing wave- and particle-like aspects of electromagnetic disturbances within the so far experimentally/observationally investigated spectrum.